Forces on an A.C.V. Executing an Unsteady Motion 



The wave resistance for steady motion in an endless tank 

 may be obtained from Eq. (29) by setting up a laterally disposed 

 array of images. The result, derived by Newman and Poole, in the 

 present notation, is 



7 7 7 



2k k • tanh(k d) |p +Q ( 



R =— | 2 e m 22 E L*2 22 , (43) 



Pg m,0 2k _ k tanh ( k d ) _ k k d . sec h 2 (k d) 

 m o m mo m 



in which u is given by Eq. (39) and w bv 



m ' m ' 



. 2 2 2 



k = w + u 



m m m 



The circular wave number, k , is the solution of 



m 



2 

 k - k k tanh(k d) = u (44) 



m m o m m v ' 



(The value of k m when m = is distinct from, and generally not 

 equal to, k Q , the fundamental wave number. ) 



IV. 3. Results 



The wave resistance of a smoothed rectangular distribution 

 moving in a tank is shown in Fig. 9. In deep water (Fig. 9a), it is 

 seen that the effect of the walls is small for B/a = 2 . For B/a > 4 

 (tank width greater than four times model width), the resistance coef- 

 ficient differs from the infinite width value by less than 0. 01. It may 

 be pointed out here that for the special case of B/a = 1 , that is, the 

 tank width equal to the nominal beam of the model, the pressure 

 carries approximately 7% of the weight of the ACV beyond the tank 

 walls. However, it can be shown that this case is mathematically 

 equivalent to a two-dimensional pressure band spanning the width of 

 the channel. 



In finite depth (Fig. 9b) the influence of the tank walls in the 

 region of unit depth Froude number is considerably greater, as was 

 shown by Newman and Poole. The drop in wave resistance (Eq. (4)) 

 at the critical speed does not depend on smoothing. Even when 

 B/a = 64 , so that the tank width is sixty-four times the model beam 

 there is a discontinuity in resistance coefficient of 0. 188 . Thus 

 steady-state experiments in this speed range are difficult. 



51 



